Problem 6. (Probabilistic Inventory Model. Total 15pts.) The Sweet Cream Dairy sells walnut fudge ice cream. Assume that historical accounting data shows that the daily Demand for ice cream can be approximated by a normal distribution with a mean of 10 gallons and a standard deviation of 5 gallons. The order delivery lead-time for restocking is 4 days.

a) Find the inventory level at which the manager should reorder if the Sweet Cream Dairy manager desires a 95% service level.

b) Assume that you order 200 gallons of ice cream each time you make an order. Use the record point you obtained in a), calculate the percentage of customer demands that cant be satisfied. (If you are not able to solve a), assume it is ROP = 50 so that you can continue work on question b).)

c) Assume that, yesterday morning, the manager of Sweet Cream placed an order of 200 gallons of the walnut fudge ice cream right after he found the inventory dropped to 50 gallons. This morning, he received a phone call from the supplier that the order he made yesterday will be delayed by 2 days due to an unexpected machine breakdown in thesuppliers factory. That is, the order will arrive 5 days after today. After the phone call, the manager checked inventory and found 8 gallons of ice cream had been sold since yesterday. Calculate the chance that the store will run out of stock at the time the order finally arrives.